Can I leave my visa stamp in my old passport to transfer it later to my new passport?

They'll certainly want to see everything transferred before you apply for the extension. In my case just transferring stamps took hours as they are seemingly very suspicious of the new UK passports which, seemingly, have a reputation for being counterfeited. Immigration were very apologetic about it but insisted they had to check. I went and had a long lunch in the middle of the process

Although there is no real hard fast rule about it I'd say you should go transfer the stamps as soon as conveniently possible 🙂

The choice is yours,